Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 036, 34 pages      arXiv:1902.05207      https://doi.org/10.3842/SIGMA.2020.036

Note on the Retarded van der Waals Potential within the Dipole Approximation

Tadahiro Miyao
Department of Mathematics, Hokkaido University, Sapporo, Japan

Received February 27, 2019, in final form April 14, 2020; Published online April 26, 2020

Abstract
We examine the dipole approximated Pauli-Fierz Hamiltonians of the nonrelativistic QED. We assume that the Coulomb potential of the nuclei together with the Coulomb interaction between the electrons can be approximated by harmonic potentials. By an exact diagonalization method, we prove that the binding energy of the two hydrogen atoms behaves as $R^{-7}$, provided that the distance between atoms $R$ is sufficiently large. We employ the Feynman's representation of the quantized radiation fields which enables us to diagonalize Hamiltonians, rigorously. Our result supports the famous conjecture by Casimir and Polder.

Key words: retarded van der Waals potential; non-relativistic QED; Pauli-Fierz Hamiltonian; dipole approximation.

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